The description of the rational Hodge structure of \(N\)  in terms of the rational Hodge structure of \(J\)  can be read off from the \(LLV\)  decomposition of the cohomology of \(N\)  given in Corollary 3.6 of [1]}. Using the branching rules described in [2]}, one computes that the rational Hodge structure of \(N\)  is given by
\(H^*(N) = \operatorname{Sym}^3U \oplus U^{\otimes 2}\langle 1 \rangle \oplus U^{\oplus 16}\langle 2 \rangle \oplus \mathbb {Q}^{\oplus 256}\langle 3 \rangle \oplus (U\otimes W)^{\oplus 2}\langle 1 \rangle .\) 
